Semantic Proof of Generalised Cut Elimination for Deep Inference System MAV

Core Concepts

Every proof in the deep inference system MAV can be normalized to a cut-free proof using a semantic model construction.

Abstract

The content presents a semantic proof of the admissibility of the cut rule and other "non-analytic" rules in the deep inference system MAV, which extends the basic system BV with additives. Key highlights: MAV is a deep inference system that extends linear logic with a self-dual non-commutative connective capturing sequential composition. Existing proofs of cut elimination for deep inference systems like MAV rely on intricate syntactic reasoning and complex termination measures. The authors develop an algebraic semantics for MAV, called MAV-algebras, and a weaker notion of MAV-frames. They show that every MAV-frame can be completed to an MAV-algebra, and that the MAV-frame constructed from normal proofs is strongly complete for MAV. This allows them to prove that every MAV-provable structure has a normal proof, avoiding the use of the cut rule and other "non-analytic" rules. The proofs are constructive and have been mechanized in the Agda proof assistant.

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A Semantic Proof of Generalised Cut Elimination for Deep Inference

by Robert Atkey... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06233.pdf

Deeper Inquiries

What are some potential applications of the deep inference system MAV and its semantics beyond proof normalization

The deep inference system MAV and its semantics have potential applications beyond proof normalization. One application could be in the field of automated reasoning and theorem proving. The semantic techniques developed in this work could be used to enhance the efficiency and accuracy of automated theorem provers by providing a more structured and robust foundation for reasoning about complex logical systems. Additionally, the insights gained from the semantic proof of generalised cut elimination could be applied to optimize proof search algorithms and improve the overall performance of automated reasoning systems.

How could the techniques developed in this work be extended to other deep inference systems or logics with self-dual connectives

The techniques developed in this work for the deep inference system MAV and its semantics could be extended to other deep inference systems or logics with self-dual connectives. By adapting the semantic proof methods and constructions used in this work, researchers could explore the properties and metatheory of a wide range of logics that incorporate self-dual connectives. This extension could lead to a deeper understanding of the relationships between different logics and provide insights into the general principles underlying proof normalization and cut elimination in various logical systems.

Are there any connections between the MAV-frame structures and models of concurrent process algebras that could be further explored

There are potential connections between MAV-frame structures and models of concurrent process algebras that could be further explored. MAV-frame structures, with their interpretation as process algebras in the context of CCS-like systems, offer a promising avenue for investigating the relationship between logical systems and concurrent computation models. By exploring the similarities and differences between MAV-frame structures and existing models of concurrent processes, researchers could potentially uncover new insights into the formalization and analysis of concurrent systems. This exploration could lead to the development of novel approaches for reasoning about concurrent processes using the framework of deep inference systems.